The Egyptians expressed all fractions as the sum of different unit fractions. The Greedy Algorithm might provide us with an efficient way of doing this.

A jigsaw where pieces only go together if the fractions are equivalent.

Is it always possible to combine two paints made up in the ratios 1:x and 1:y and turn them into paint made up in the ratio a:b ? Can you find an efficent way of doing this?

A decorator can buy pink paint from two manufacturers. What is the least number he would need of each type in order to produce different shades of pink.

Can all unit fractions be written as the sum of two unit fractions?

The number 2.525252525252.... can be written as a fraction. What is the sum of the denominator and numerator?

Can you see how to build a harmonic triangle? Can you work out the next two rows?

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

Is there an efficient way to work out how many factors a large number has?

Do you know a quick way to check if a number is a multiple of two? How about three, four or six?

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

Can you find the area of a parallelogram defined by two vectors?

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

A game for 2 or more people, based on the traditional card game Rummy. Players aim to make two `tricks', where each trick has to consist of a picture of a shape, a name that describes that shape, and. . . .

How many winning lines can you make in a three-dimensional version of noughts and crosses?

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?

Different combinations of the weights available allow you to make different totals. Which totals can you make?

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

Water freezes at 0°Celsius (32°Fahrenheit) and boils at 100°C (212°Fahrenheit). Is there a temperature at which Celsius and Fahrenheit readings are the same?

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

Some people offer advice on how to win at games of chance, or how to influence probability in your favour. Can you decide whether advice is good or not?

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

If you move the tiles around, can you make squares with different coloured edges?

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?

Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

How many different symmetrical shapes can you make by shading triangles or squares?

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...